N-D Test Functions T¶

class TestTubeHolder(dimensions=2)¶

TestTubeHolder objective function.

This class defines the TestTubeHolder global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{TestTubeHolder}}(x) = - 4 \left | {e^{\left|{\cos \left(\frac{1}{200} x_{1}^{2} + \frac{1}{200} x_{2}^{2}\right)} \right|}\sin\left(x_{1}\right) \cos\left(x_{2}\right)}\right|$

with $x_i \in [-10, 10]$ for $i = 1, 2$ .

Two-dimensional TestTubeHolder function

Global optimum: $f(x) = -10.872299901558$ for $x= [-\pi/2, 0]$

Mishra, S. Global Optimization by Differential Evolution and Particle Swarm Methods: Evaluation on Some Benchmark Functions. Munich Personal RePEc Archive, 2006, 1005

class ThreeHumpCamel(dimensions=2)¶

Three Hump Camel objective function.

This class defines the Three Hump Camel global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{ThreeHumpCamel}}(x) = 2x_1^2 - 1.05x_1^4 + \frac{x_1^6}{6} + x_1x_2 + x_2^2$

with $x_i \in [-5, 5]$ for $i = 1, 2$ .

Two-dimensional ThreeHumpCamel function

Global optimum: $f(x) = 0$ for $x = [0, 0]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Thurber(dimensions=7)¶

Thurber objective function.

http://www.itl.nist.gov/div898/strd/nls/data/thurber.shtml

class Treccani(dimensions=2)¶

Treccani objective function.

This class defines the Treccani global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Treccani}}(x) = x_1^4 + 4x_1^3 + 4x_1^2 + x_2^2$

with $x_i \in [-5, 5]$ for $i = 1, 2$ .

Two-dimensional Treccani function

Global optimum: $f(x) = 0$ for $x = [-2, 0]$ or $x = [0, 0]$ .

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Trefethen(dimensions=2)¶

Trefethen objective function.

This class defines the Trefethen global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Trefethen}}(x) = 0.25 x_{1}^{2} + 0.25 x_{2}^{2} + e^{\sin\left(50 x_{1}\right)} - \sin\left(10 x_{1} + 10 x_{2}\right) + \sin\left(60 e^{x_{2}}\right) + \sin\left[70 \sin\left(x_{1}\right)\right] + \sin\left[\sin\left(80 x_{2}\right)\right]$

with $x_i \in [-10, 10]$ for $i = 1, 2$ .

Two-dimensional Trefethen function

Global optimum: $f(x) = -3.3068686474$ for $x = [-0.02440307923, 0.2106124261]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Trid(dimensions=6)¶

Trid objective function.

This class defines the Trid global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Trid}}(x) = \sum_{i=1}^{n} (x_i - 1)^2 - \sum_{i=2}^{n} x_i x_{i-1}$

Here, $n$ represents the number of dimensions and $x_i \in [-20, 20]$ for $i = 1, ..., 6$ .

Two-dimensional Trid function

Global optimum: $f(x) = -50$ for $x = [6, 10, 12, 12, 10, 6]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

Jamil#150, starting index of second summation term should be 2.

class TridiagonalMatrix(dimensions=2)¶: TridiagonalMatrix objective function.

Two-dimensional TridiagonalMatrix function

class Trigonometric01(dimensions=2)¶

Trigonometric 1 objective function.

This class defines the Trigonometric 1 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Trigonometric01}}(x) = \sum_{i=1}^{n} \left [n - \sum_{j=1}^{n} \cos(x_j) + i \left(1 - cos(x_i) - sin(x_i) \right ) \right]^2$

Here, $n$ represents the number of dimensions and $x_i \in [0, \pi]$ for $i = 1, ..., n$ .

Two-dimensional Trigonometric01 function

Global optimum: $f(x) = 0$ for $x_i = 0$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

Todo

equaiton uncertain here. Is it just supposed to be the cos term in the inner sum, or the whole of the second line in Jamil #153.

class Trigonometric02(dimensions=2)¶

Trigonometric 2 objective function.

This class defines the Trigonometric 2 global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Trigonometric2}}(x) = 1 + \sum_{i=1}^{n} 8 \sin^2 \left[7(x_i - 0.9)^2 \right] + 6 \sin^2 \left[14(x_i - 0.9)^2 \right] + (x_i - 0.9)^2$

Here, $n$ represents the number of dimensions and $x_i \in [-500, 500]$ for $i = 1, ..., n$ .

Two-dimensional Trigonometric02 function

Global optimum: $f(x) = 1$ for $x_i = 0.9$ for $i = 1, ..., n$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Tripod(dimensions=2)¶

Tripod objective function.

This class defines the Tripod global optimization problem. This is a multimodal minimization problem defined as follows:

$f_{\text{Tripod}}(x) = p(x_2) \left[1 + p(x_1) \right] + \lvert x_1 + 50p(x_2) \left[1 - 2p(x_1) \right] \rvert + \lvert x_2 + 50\left[1 - 2p(x_2)\right] \rvert$

with $x_i \in [-100, 100]$ for $i = 1, 2$ .

Two-dimensional Tripod function

Global optimum: $f(x) = 0$ for $x = [0, -50]$

Jamil, M. & Yang, X.-S. A Literature Survey of Benchmark Functions For Global Optimization Problems Int. Journal of Mathematical Modelling and Numerical Optimisation, 2013, 4, 150-194.

class Tsoulos(dimensions=2)¶: Tsoulos objective function.

Two-dimensional Tsoulos function

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